Question

which expression is equivalent to the following complex fraction?\\(\frac{\frac{2}{x}-\frac{4}{y}}{\frac{-5}{y}+\frac{3}{x}}\\)\\(\circ\frac{3y + 5x}{2(y - 2x)}\\)\\(\circ\frac{2(y - 2x)}{3y - 5x}\\)\\(\circ\frac{2(y - 2x)(3y - 5x)}{x^{2}y^{2}}\\)\\(\circ\frac{x^{2}y^{2}}{}\\)

Explanation:

Step1: Simplify numerator

$$\frac{2}{x} - \frac{4}{y} = \frac{2y - 4x}{xy} = \frac{2(y - 2x)}{xy}$$

Step2: Simplify denominator

$$\frac{-5}{y} + \frac{3}{x} = \frac{-5x + 3y}{xy} = \frac{3y - 5x}{xy}$$

Step3: Divide numerator by denominator

$$\frac{\frac{2(y - 2x)}{xy}}{\frac{3y - 5x}{xy}} = \frac{2(y - 2x)}{xy} \times \frac{xy}{3y - 5x}$$

Step4: Cancel common terms

$$\frac{2(y - 2x)}{3y - 5x}$$

Answer:

$\boldsymbol{\frac{2(y-2x)}{3y-5x}}$ (the second option)